Download Simulation of AC Electrical Machines Behaviour Using Discrete

Simulation of AC Electrical Machines Behaviour Using
Discrete Event System Simulator
Laurent Capocchi, Dominique Federici, Humberto Henao, Gerard-André
Capolino
To cite this version:
Laurent Capocchi, Dominique Federici, Humberto Henao, Gerard-André Capolino. Simulation
of AC Electrical Machines Behaviour Using Discrete Event System Simulator. 2007 IEEE
International Symposium on Industrial Electronics, Jun 2007, Vigo, Spain. 14 (7), pp.945-970,
2007.
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Simulation of AC Electrical Machines Behaviour
Using Discrete Event System Simulator
L. Capocchi
University of Corsica
UMR CNRS 6134
Quartier Grossetti, BP 52
20250 Corte - FRANCE
[email protected]
H. Henao
G. A. Capolino
D. Federici
University
of
Picardie
University of Picardie
University of Corsica
Dpt. of Electrical Eng.
Dpt. of Electrical Eng.
UMR CNRS 6134
33, rue St Leu
33, rue St Leu
Quartier Grossetti, BP 52
80039
AmiensFRANCE
80039 Amiens - FRANCE
20250 Corte - FRANCE
[email protected] [email protected]
[email protected]
Abstract— This paper deals with the adaptation of AC electrical machine models for discrete event system simulator. The
formalism chosen is DEVS (Discrete Event system Specification)
which has been adapted recently for hybrid system simulation.
The software PowerDEVS is close to MATLAB/Simulink(C)
but without any toolbox adapted to power systems. As always, the model which is basically based on system of nonlinear differential equations has to be integrated to compute
the output variables corresponding to any deterministic or
stochastic inputs. In this way, the integrator atomic model
has been optimized to perform many simulations with realistic
CPU time. Results obtained with PowerDEVS and optimized
graphical model have been validated by the same simulations
using MATLAB/Simulink(C) as found in the classical literature.
I. I NTRODUCTION
The simulation programs for electrical power systems have
been used since the beginning of the 60s. There are more
than hundred of them and this is event difficult to have an
exhaustive list and/or a classification. However, some of them
are well known and have been described in the literature such
as EMTP, ECAP and NETOMAC which have been initiated
in the late 60s. Some more recent package such as SPICE (its
adaptation to power systems) or MATLAB (its power systems
simulator) have been successfully used for many applications.
Others such as Dymola, Omsim, EUROSTAG can be classified
in software used for power systems simulation [1]. Many of
them are adapted to simulation a model defined by sets of
ordinary differential equations (ODE) which are related generally to continuous systems under textual or block-diagram
representations. During the last twenty years, many efforts
have been made in order to give the user modular modelling
environments with easy user-based interfaces for transparent
and automatic simulation. Nowadays, all the available programs propose graphical interfaces to simplify the modelling
process of any physical systems. However, these interfaces
are more and more complex and can only be fully exploited
by experts of the studied domain. Moreover, the simulation
algorithms proposed within the programs are mainly based
on numerical integration methods such as Runge-Kutta, Euler,
Adams and so on. These methods are specific in the sense that
they are all based on a classical discretization resulting in a
1-4244-0755-9/07/$20.00 '2007 IEEE
discrete time simulation model that could require much more
execution time.
So far, electrical machine digital simulation has been under
focus for the last fourty years and many of the previous
software have been tested for this purpose. The result of this
extended research is that many of these packages have their
own power systems library inside. However, the approach
used is always the same since the electrical machine has
to be considered as a continuous system which has to be
discretized to be digitally simulated in any of these packages.
Two different ways are available, the first one being related to
direct discretization of non-linear differential equations and
the second being indirect by the interpretation of internal
model with a circuit-oriented approach. For any simple threephase AC machine in symmetrical conditions, the problem
is very simple to be solved and the two methods are almost
equivalent in term of complexity. However, when the windings
are not symmetrical and/or if any other fault did occur in
the electromechanical interface the circuit-oriented approach
has been proved to be more flexible in term of physical
interpretation. Since a fault can be considered as a discrete
event, it is necessary to define a suitable model corresponding
with its nature for any complex system having one (or several)
electrical machine inside such as wind turbine or microhydro generators. For this purpose, a new formulation of
AC electrical machine models is mandatory to export them
in the main discrete event simulators developed for different
applications.
The aim of this paper is to present a new modelling and
simulation approach of a complex power systems using a
DEVS-based (Discrete EVent system Specification) graphical
environment. The DEVS formalism [2] has been defined thirty
years ago to allow the specification of discrete event systems.
It provides a way to define complex models in a hierarchical
and modular way. This environment allows the implementation
of numerical simulation algorithms based on a discretization
which is not linked to time anymore but to space on state
variables if the system is continuous. Many years ago, DEVS
was upgraded in order to permit the modelling of continuous
systems and two main formalisms were developed. In GDEVS
(Generalized Discrete EVent Specification) [3], the trajectories
are organized through piecewise polynomial segments. In
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the literature, the key contribution of GDEVS is considered
as its ability to develop uniform discrete event executable
specifications for hybrid dynamic systems. Moreover, discrete
event systems including DEVS and GDEVS are simulated at
high speed on a host computer because of significant changes
in the system. By speaking again of ODE efficient numerical simulation, another DEVS-based method called quantized
systems was proposed by Zeigler [4]. In this approach, the
time discretization is replaced by the state quantization and a
DEVS simulation model is obtained instead of a discrete time
one. This idea was reformulated and formalized by Kofman
[5] as a simulation method for general ODE in where the
quantized state system (QSS) method was defined. In [6], the
author shows that from the computational cost point of view,
the QSS method can reduce the number of iterations. Many
examples like block-oriented DEVS simulation of a RLC
circuit have been recently presented [6] but the configuration
of the QSS integrator model is not explicit. Another example
of the circuit simulation based on bond-graph translation was
also presented [7].
In this paper, an extension of the use of the QSS method for
the simulation of differential algebraic equations (DAE) which
describe any electrical power system. According to [6], the
proposed approach is based on block-by-block translation from
a block-diagram resulting from the DAE systems. Each step of
the modelling process is described and every approach is discussed. In order to achieve the proposed modelling approach,
the package PowerDEVS [8] is used and it is adapted to a more
complex system than originally designed for. PowerDEVS
is a DEVS-based graphical environment for hybrid system
modelling and simulation. It proposes several QSS methods
which provide a quantization of the state variables to obtain a
discrete event approximation of the continuous system.
The paper is organized in four sections. The section 2
presents the background and shows the DEVS formalism
with the DEVS-based modelling and simulation software
PowerDEVS. This section briefly describes the QSS methods
proposed in PowerDEVS. In section 3, the use of PowerDEVS
environment for electrical power systems modelling and simulation is proposed. Therefore, the case of a simple electrical
circuit with its differential equations is presented. Then, the
PowerDEVS simulation is given and the results are analyzed.
Finally, the section 4 gives some conclusions and directives
for a future direction of research in the usage of discrete event
simulation for power systems fault detection.
II. R ELATED W ORKS
A. DEVS formalism
The DEVS formalism was introduced by Zeigler in the
late 70s [9], [10]. It is based on systems theory, allowing
a hierarchical and modular way to model the discrete event
systems. A system or a model is called modular, if its input and
output ports interact with its environment. In DEVS, a model
is seen as a “black box” receiving and sending messages on
its input and output ports respectively. Two types of models
are defined, atomic models and coupled models, representing
respectively the behaviour and the internal structure of a model
element.
Fig. 1.
DEVS atomic model
Figure 1 represents an AM atomic model with its output data
Y calculated according to input data X. The atomic model has
a state variable S that can be observed during the simulation.
When an external event occurs, the model state is modified
depending of the following functions:
• δext : external transition function
• λ: output function
• δint : internal transition function
• ta : time advance function
The coupled models are defined by a set of sub-models (atomic
and/or coupled) and express the internal structure of the system
sub-elements thanks to the coupling definition between the
sub-models.
Fig. 2.
DEVS coupled model
Figure 2 shows an example of a coupled model hierarchical
structure. The coupled model CM0 has an input port IN and
two output ports OUT0 and OUT1 . It contains the atomic
sub-models AM0 , AM1 and also the coupled model CM1 .
This latter is constitued by the atomic models AM2 , AM3
and AM4 . A coupled model is specified by the list of its
components (AM0 , AM1 ,AM2 , AM3 , AM4 and CM1 ), the list of
its internal couplings (AM0 → CM1 and AM1 → CM1 ), the list
of the external input couplings (IN → AM0 and IN → AM1 ),
the list of the external output couplings (CM1 → OUT0 and
CM1 → OUT1 ) and the list of the sub-model influence (CM1 =
{AM0 , AM1 } or CM1 and influenced by AM0 and AM1 ).
The DEVS formalism is mainly used for the description of
discrete event systems. It constitutes a powerful modelling and
simulation tool permitting a system modelling with several
levels of description as well as the definition of the model
behaviors. It provides a simulator for each model automatically
and establishes a distinction between a system modelling
and a system simulation. Each atomic model is associated
with a simulator which manage the component behavior.
Each coupled model is associated with a coordinator which
synchronizes temporally its components.
B. PowerDEVS Environment
Some of DEVS simulation tools as AToM3 [11], DiamSim [12] and PowerDEVS [8] include graphical interface
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and advanced simulation features. These tools were developed
before the creation of discrete event methods for numerical
integration of ODE, excepting PowerDEVS. This last software
is a DEVS-based integrated tool for modelling and simulation
of continuous systems. It is composed by four separate programs [8]:
• The model and atomic editor
• The structure generator and the pre-processor
PowerDEVS allows building or managing atomic DEVS models or libraries. These models can be graphically coupled in
a hierarchical way to create more complex systems. Atomic
and coupled models can compose libraries, facilitating the application of all the defined components. For the power system
simulation, the PowerDEVS “Continuous” library is a well
adapted tool to represent the complexity of electromagnetic
devices. As it is shown in Figure 3, this library composed
by most of the components that are used to build the blockschemes corresponding to the ODE of continuous systems.
Fig. 3.
PowerDEVS “Continuous” Library.
This “Continuous” library proposes an integrator atomic
model to perform the ODE integration with optimized state
quantization methods [6].
C. State Quantization-Based Methods
The basis of the state quantization-based methods comes
from the idea that, it is possible to obtain an ODE numerical
simulation by a discrete representation of state variables, instead of a discrete representation in time. This idea introduced
initially by Zeigler was reformulated defining in a formal way
a quantized state system (QSS) for a first order approximation,
to formalize a new numerical integration approach [5]. A QSS
is based on a hysteresis quantization method where:
• The quantized variables have piecewise trajectories
• The state variable have also piecewise trajectories
• The state variables have continuous piecewise trajectories
Figure 4 shows a typical quantization function q(t) with
uniform quantization intervals, obtained with a hysteresis window ε. With this quantization method, the stability properties
of the original continuous system are conserved. When the
window ε is chosen to be equal to the quantum q, there is a
reduction in the number of steps performed by the hysteresis
algorithm and a consequent reduction of the computational
costs. With this method the obtained accuracy depends on the
quantum q, which is related to the number of hysteresis cycles.
Then a good accuracy cannot be obtained without increasing
significantly the computational cost.
Fig. 4.
Quantized function with hysteresis.
A quantized state system for a second order approximation
(QSS2) [13], [14] has been proposed in order to obtain
less number of hysteresis cycles with respect to the QSS
method for the same quantum q. The QSS2 method uses first
order quantizers, conducing to piecewise linear trajectories in
the quantized variables. The development of a third order
approximation QSS3 is introduced in [15]. With this new
approach, the accuracy can be improved and the quantization
choice is not critical as with QSS2. The obtained results show
that this last method is an efficient algorithm for accurate
numerical integration of discontinuous systems.
III. DEVS S IMULATION OF P OWER S YSTEMS
Electrical power systems models are essentially build
around resistances (R), inductances (L), capacitors (C) and
voltage sources (v) and current sources (i). The equations relating voltages and currents in the passive elements characterized
by R, L and C, describe the physical behaviour of electrical
devices with respect to the energy converted into heat (R), the
energy stored in both inductive (L) and capacitive (C). In this
way, the complete model will be translated into a continuous
system described by a set of DAE.
A. Induction Machine Stator Side
The proposed circuit is a symmetrical three-phase magnetically linked inductive circuit which is the simple low frequency
representation of any AC machine stator windings. In order
to obtain a linear circuit-oriented model of this device, the
passive elements are used taking into account the resistive and
inductive effects (Fig. 5). The capacitive effects are neglected
in order to remain within the low frequency analysis range.
Fig. 5.
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Symmetrical three-phase circuit
The elements of the equivalent circuit are given by:
• rs : phase resistance,
• Ls : phase self-inductance,
• Lms : mutual inductances between phases.
The power supply is given by the three-phase symmetrical
voltages vas (t), vbs (t) and vcs (t) which are modelled as sinusoidal time functions:
⎧
⎨ vas (t) = Vm sin(2π f t)
vbs (t) = Vm sin(2π f t − 2π
(1)
3 )
⎩
4π
vcs (t) = Vm sin(2π f t − 3 )
√
√ 2 , Um being the line-to-line voltage and f
with Vm = Um.
3
the suply frequency.
Applying the electric circuit laws to the circuit of Fig 5, it
a set of equations can be obtained:
⎧
L
⎨ vas = rs .ias + Ls dtd ias − 2ms dtd ibs + dtd ics v = rs .ibs + Ls dtd ibs − L2ms dtd ias + dtd ics
⎩ bs
vcs = rs .ics + Ls dtd ics − L2ms dtd ias + dtd ibs
(2)
The set of formulas 2 shows a DAE system where the
continuous variables ias , ibs and ics represent the state variables
(phase currents) to be computed. Moreover, this system has to
be translated in the PowerDEVS environment.
The PowerDEVS Continuous library has that element and
its implementation requires the determination of weighting
coefficients Ki,i∈{0,1,2,3} of each term of the sum. Since a
system of three equations is provide with one equation per
phase x (x ∈ {a, b, c}), three adders Sumi,i∈{1,2,3} with four
inputs will be necessary with:
1
• the input voltage vxs with a weighting coeff. of K0 = L ,
rSs
• the stator current ixs with a weighting coeff. of K1 = − L ,
S
• the two derivatives of the two stator currents with the
Lms
weighting coeff. of K2 = K3 = 2.L
.
S
It is necessary to use a coupled model called “stator” which
includes three adders. As it is shown (Fig. 8), the three-phase
stator model has three inputs and three outputs. The three
inputs allow the acquisition of the sinusoidal input voltages
vxs . The three outputs allow the computation of the three stator
currents. The output of each adder provides the stator current
derivatives and they are used to compute the other stator
currents being injected in the last two inputs of the neighbour
adders. However, these derivatives introduce a numeric noise
and this can be drastically reduced by implementing one
low-pass filter LPi,i∈{0,1,2} in each connexion. Each filter is
represented by one black-box in each coupled model (Fig. 8).
B. PowerDEVS Simulation
The proposed circuit can be described by block-diagrams
with algebraic loops. In this way, the DAE system must be
expressed as a function of state variables and their derivatives:
⎧
Lms d
1
d
d
⎪
⎨ dt ias = Ls vas − rs ias + 2 dt ibs + dt ics Lms d
1
d
d
(3)
dt ibs = Ls vbs − rs ibs + 2 dt ias + dt ics
⎪
L
⎩ d i = 1 v − r i + ms d i + d i
cs
s cs
dt cs
Ls
2
dt as
dt bs
For each state variable, a block-diagram has to be described
coupling a voltage source with an adder and an integrator
(Fig. 6). Then, two types of algebraic loops are presents for
the state variable computation. The first one is related to the
initial state variable and the second one shows the mutual
effect between all the state variables.
Fig. 6.
Stator Block Diagram
Fig. 7.
Low-Pass Coupled Model
A simple block-scheme representation of each low-pass
filter is very classical and can be easily provided (Fig. 7). It
is composed by one adder and one integrator interconnected
within a closed-loop system. The characteristics of the lowpass filters depend on the stator winding time constant. Indeed,
if the equivalent time constant is τ = RL , the reverse of the adder
weighting coefficient must be very low in front of τ (10 times
less at least).
Fig. 8.
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Stator Coupled Model with Low-pass Filters
The complete system will be simulated considering a set of
sinusoidal input voltages Vxs . The simulation of the complete
three-phase system will be tested using different integration
methods with the quantized state systems (QSS, QSS2, QSS3)
and the quantum dq within the Integrator atomic models 1, 2
and 3 (Fig. 8).
dq
0.1
0.01
0.001
0.001
QSS
Time
Event
1.485
163
2.632
2000
10.969
20301
96.652 203381
QSS2
Time Event
1.255
86
1.198
322
1.289 1040
2.549 3287
QSS3
Time Event
0.857
71
0.922
199
0.925
440
0.938 1010
TABLE II
S IMULATION RESULTS
C. Simulation Results and Performance Analysis
The numerical parameters used to simulate the set of
equations (3) are summarized in Table I. These values allow
to compute the weighting coefficients Ki,i∈{0,1,2,3} of each
adder which are part of the induction machine stator model.
of the adders included
The weighting coefficients Ki,i∈{0,1}
in the low-pass filters are computed considering the fact that
τ = RL 0.03s. As it has been already said previously, the
reverse coefficients must be much lower than to 0.03s. In this
way, the final choice has been K0 = −K1 = 10000. Moreover,
each integrator within the filters uses the QSS2 method with
the quantum dq = 0.001 and without any initial condition.
Line-to-line voltage (Um )
Suply frequency ( f )
Stator resistance (rs )
Stator inductance (Ls )
Stator magnetic inductance (Lms )
K0 = L1
S
K1 = LrS
S
Lms
K2 = 2.L
S
K3 = K2
K0 = −K1
380V
50Hz
79.13 Ω
2.83H
2.2H
0.353357
−27.96110
0.388690
0.388690
10000
TABLE I
VALUES OF ALL PARAMETERS
As it is shown in the complete block-scheme (Fig. 9),
the output stator currents are observed by connecting
QuickScopei,i∈{1,2,3} atomic models from the “Sinks” library
to the three outputs ias (t), ibs (t) and ics (t) of the stator coupled
model. The complete system was simulated with four different
values of the quantum dq which influences at the level of the
stator integrators. The effect of the quantization methods QSS,
QSS2 and QSS3 has been also examined.
These data are available using the functions “Time Counter”
and “Event Counter” atomic models from the PowerDEVS
“Sinks” library. The previous results show that the number of
events and the simulation time are in reverse proportion to the
quantum dq for any method used. It is due to the fact that the
smaller this quantum is, the larger the number of events will
be. Therefore, the simulation time will be larger with small
quantum.
The accuracy of the results has been demonstrated using a
simple steady state simulation (Fig. 10). The variable plotted
is time evolution of the stator current ias (t) in the first
phase considering quantum dq for the QSS, QSS2 and QSS3
methods. The proposed simulation does not show the transient
which clearly disappears after regime which disappears after
10 periods (with a period T = 0.02s) to remain in steady state.
It is more than clear that the quantum has a large influence
of the result accuracy. A zoom is performed during the steady
state (Fig. 10.b) in between the time 2.5s and 2.9s for quantum
values ranging from 0.1 to 0.0001 by steps of 10−1 . The
expected solution is computed by integration of the set of
ODE with classical Euler method with a time step of 0.001ms.
When the quantization step is 0.1, it can be observed that
the step-by-step solution is exact compared with the solutions
given for smaller quantum. If a linear interpolation is used, the
numerical error between this first solution and the other with
smaller quantum is very important and it is not acceptable. On
the other hand, when the quantization step reaches the value
0.001, the error using a linear interpolation is smaller than in
the previous case and the numerical error becomes acceptable.
When the quantum decreases, there is a natural increase on
the computation time. The QSS3 method allows an interesting
simulation time depending in the quantum choice. The QSS2
method is not as fast as the QSS3 method (rate of 21 with
QSS3 for dq=0.0001) but is less restrictive.
IV. C ONCLUSION AND F UTURE D IRECTIONS
Fig. 9.
Experimental framework
For each simulation of the system during t f = 0.4s, the
simulation time (in second) and the number of events obtained
under an Intel-Pentium M processor 1.70GHz with 1Go of
RAM are reported (Table II).
In this paper, it has been shown that the PowerDEVS environment allows the modelling and the simulation of complex
electrical power systems. The proposed implementation for
the induction machine stator model has permitted to test the
QSS quantization methods and to appreciate how efficient
they can be in term of computational cost and efficiency. The
QSS2 method gives the better results compared to its former
version QSS. Indeed, the usage of QSS2 within the integration
procedure used in each phase of the stator model permits
to converge on the expected solution. The QSS3 method is
faster than QSS2 but the choice of a valid quantum is very
critical. It has been shown that the insertion of a low-pass filter
1228
connected to the output is necessary since it corresponds with
the derivative of each stator current in the three phases.
(a) with QSS
accuracy with the DEVS approach is not greatly affected
by the quantum value and then, the simulation time can be
minimized using high order interpolation methods. Moreover,
the simple DEVS approach shows a performance superior to
the complex implicit, high order and variable step methods.
This first approach of the three-phase induction machine stator
model has allowed to perform a simple and efficient model of
three-phase AC electrical machines. Indeed, the choice of the
QSS quantized method within this complete model is based
on the present paper results. The final objective will require
the definition of effective fault models from the experiments
achieved on a laboratory test-bench or on a real industrial
system. Once the AC machine models will be established
and simulated in PowerDEVS, the complete model will be
transformed into BFS-DEVS (Behavioral Fault Simulation for
DEVS) [16] in order to build a fault simulator based on the
concurrent and comparative simulation approach.
R EFERENCES
(b) with QSS2
(c) with QSS3
Fig. 10.
Simulation results for ias (t) signal.
This process is used to reduce the delay introduced by
the derivation and it is known as a stabilization of the
final solution. Moreover, this paper presents the adaptation
of the low-pass filter time constant to reduce the simulation
time. This approach is usual to that kind of systems but
it has never been tested in the PowerDEVS environment.
In a short-term prospect, it is needed to extend the stator
model to the complete three-phase squirrel-cage or woundrotor induction machine in order to perform effective fault
simulation as a discrete event phenomenon. The obtained
results with PowerDEVS show some advantages compared
with other simulation packages which use discrete time integration methods as MATLAB/SIMULINK(C). Firstly, the
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